Abstract

When solutions of the stationary Schrödinger equation in a half-space belong to the
weighted Lebesgue classes, we give integral representations of them, which imply known representation
theorems of classical harmonic functions in a half-space.

1. Introduction and Results

Let and be the sets of all real numbers and of all positive real numbers, respectively. Let denote the -dimensional Euclidean space with points , where and . The unit sphere and the upper half unit sphere in are denoted by and , respectively. The boundary and closure of an open set of are denoted by and , respectively. The upper half space is the set , whose boundary is .

We identify with and with , writing typical points as , where and putting

For and , let denote the open ball with center at and radius in . , where is the origin of . For a set , , we denote and by and , respectively. By we denote . We denote by the -dimensional volume elements induced by the Euclidean metric on .

Let denote the class of nonnegative radial potentials , that is, , , such that with some if and with if or .

This paper is devoted to the stationary Schrödinger equation:
where , is the Laplace operator and . Note that solutions of (2) are (classical) harmonic functions in the case . Under these assumptions the operator can be extended in the usual way from the space to an essentially self-adjoint operator on (see [1]). We will denote it by as well. This last one has a Green function . Here is positive on and its inner normal derivative . We denote this derivative by , which is called the Poisson -kernel with respect to . If and are denoted by the Green function and Poisson kernel of the Laplace operator in , respectively, then
where , and is the area of the unit sphere in .

Let be a continuous function on . We say that is a solution of the Dirichlet problem for the stationary Schrödinger operator on with , if
Note that is a solution of the classical Dirichlet problem for the Laplace operator on with in the case .

Let be a Laplace-Beltrami operator (spherical part of the Laplace) on the unit sphere. It is known (see, e.g., [2, page 41]) that the eigenvalue problem
has the eigenvalues , where . Corresponding eigenfunctions are denoted by , where is the multiplicity of . We norm the eigenfunctions in and .

Let and stand, respectively, for the increasing and nonincreasing, as , solutions of the equation:
normalized under the condition .

We shall also consider the class , consisting of the potentials such that there exists a finite limit ; moreover, . If , then solutions of (2) are continuous (see [3]).

In the rest of paper, we assume that and we shall suppress this assumption for simplicity. Further, we use the standard notations , , and is the integer part of and , where is a positive real number.

Denote

Remark 1. in the case , where .

It is known (see [4]) that in the case under consideration the solutions to (6) have the asymptotics
where and are some positive constants.

If , it is known that the following expansion for the Green function (see [5, Chapter 11], [6]):
where and is its Wronskian. The series converges uniformly if either or , where .

For a nonnegative integer and two points , we define a modified Green function:
where

If the generalized Poisson kernel with respect to is defined by
then we have and coincides with ones in Finkelstein and Scheinberg [7], Siegel and Talvila [8], Deng [9], Qiao and Deng [10], and Qiao [11] (see [5, Chapter 11]).

Put
where is a continuous function on .

For real numbers , we denote the class of all measurable functions satisfying the following inequality:
and the class consists of all measurable functions satisfying

We will also consider the class of all continuous functions , which is the solution of (2), with and , is denoted by .

Remark 2. If and , then (14) and (15) are equivalent to
respectively, from Remark 1 and (14), which yield that is equivalent to in the notation of [9].Let us recall the classical case . If is harmonic in , continuous on , and , then there exists a constant such that (see [12])
where .

Deng (see [9]) has constructed a similar representation to (17) for , which is the integral with a modified Poisson kernel derived by subtracting some special harmonic polynomials from . By virtue of this modified Poisson kernel, Qiao (see [11, 13]) and Qiao and Deng (see [14–18]) have constructed different integral representations for harmonic functions of finite order and infinite order.

Especially, Su (see [6, 19]) recently writes solutions to the half-space Dirichlet problem with respect to the stationary Schrödinger operator . Now we state our main results as follows.

Theorem 3. If , then .

Theorem 4. If , is an integer such that , and then the following properties hold.(I) If , then there exists a constant such that
for .(II) If , then we have , where
vanishing continuously on , and are constants.

2. Lemmas

The following Lemma plays an important role in our discussions, which is due to Levin and Kheyfits (see [5, page 356]).

Lemma 5. If and is a solution of (2) on a domain containing , then
where

Lemma 6 (see [6, Corollary 1.6]). If is a continuous function on satisfying
then is a solution of the Dirichlet problem for on with and

Lemma 7 (see [6, Lemma 2.1] or [20, Theorem 1]). If is a solution of (2) on satisfying
then (19) holds.

Acknowledgments

The author wishes to expresseses his appreciation to Professor Jingben Yin for very useful conversations related to this problem. He is grateful to the referee for her or his careful reading and helpful suggestions which led to the improvement of the original paper. This work is supported by the National Natural Science Foundation of China under Grant 11226093.